Failing to test the assumption that error is normally distributed and has constant variance between treatments. These assumptions aren't always tested, thus least-squares model fitting is probably often used when it is actually inappropriate.
and @DikranMarsupial's subsequent comment:
If the data are heteroscedastic you can end up with very innacurate out of sample predictions because the regression model will try too hard to minimise the error on samples in areas with high variance and not hard enough on samples from areas of low variance. This means you can end up with a very badly biased model. It also means that the error bars on the predictions will be wrong.
I am worried about my regression: I used "rlm" with weights. My questions are:
Please give me some pointers about how to do goodness-of-fit and residual diagnosis for rlm in R? Any good tutorials/examples, etc?
How to do the residual diagnosis etc. when regression weights are used?
My end goal was to get the predictions $\hat y$... and I am less concerned about the bias or efficiency of the betas...
Do my predictions $\hat y$ get messed up if there are heteroskedasticity, non-normality, etc?
i.e. what's the relationship between efficiency/unbiasedness and good out-sample prediction?